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Math Help - Derivative of a constant complex function

  1. #1
    Ant
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    Derivative of a constant complex function

    Let f be an analytic function on a disc D.

    Is it true that:

    f is constant \iff  f'(z) =0  \forall z \in D

    with ' denoting the derivative wrt to complex variable z. Or do we have to include some conditions on the partial derivatives wrt x,y (where Re(z)=x and Im(z)=y) ?

    Thanks for anyhelp!
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  2. #2
    MHF Contributor
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    Re: Derivative of a constant complex function

    Hey Ant.

    This is true and you can use the fact that all analytic functions in a region are infinitely differentiable with the caveat that differentiating a constant gives 0.
    Thanks from Ant
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  3. #3
    Ant
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    Re: Derivative of a constant complex function

    Thanks.

    Do we also have the following:

     f'(z)=0 \iff both partial derivatives with respect to x and y are also zero.


    (I think perhaps the above is false and only the  \Rightarrow direction holds??)


    Thanks!
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  4. #4
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    Re: Derivative of a constant complex function

    Yes we should have and you could use the Cauchy-Riemann equations to show this.
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